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Fig. 7.4. The bifurcation diagram of equilibrium solutions: — stable branch,--unstable branch.

Fig. 7.4. The bifurcation diagram of equilibrium solutions: — stable branch,--unstable branch.

It is interesting that this value is constant for a given chain of any length (it follows from Eq. (5.3) if q = const); it is determined only by characters of the energy exchange between the system and the environment and does not depend on the internal system parameters. In Fig. 7.4 a so-called "bifurcation diagram" is shown.

We can see that when the value of inflow q passes over the first critical number q1 = di d2/ a1 then the equilibrium (a) loses its stability, but the new stable equilibrium (b) arises. The solution is branched; instead of one stable branch, the two branches appear. If in the vicinity of a critical number the system remains on the old branch then s = di Nip (a) + d2N2 > q. In order to return to the minimal value of s* = q, the system has to be reconstructed and the chain has to increase its length from one to two links. Moreover, the further growth of q does not increase the value of the first level: it remains constant and all the increments of q are spent in maintaining the second level, the biomass of which is growing with the growth of q. Calculating the total biomass of the chain:

(c) N1 = Ni + Ni + Ni = [q(ai + «2)/(«id3 + «2di)] + d - d2)/a2, we see that the total biomass increases with the growth of q (see Fig. 7.5).

Let us return to Eq. (5.3), from which follows that N* = q/d, and if N* ! max then d! min. It is obvious that the dk /T can be considered as a specific (per one unit of biomass) internal production of entropy at the kth level. Then d/T = (i/T)^\=i dkpk is the mean entropy production, averaged over the whole chain. Therefore, we can formulate the following statement: if the mean specific value of the internal entropy production in a given chain tends to its possible minimum then the total chain biomass tends to its possible maximum. The proof is based on the statement proven earlier that the value of s decreases along the trajectories towards a stable equilibrium. Indeed, since s = dN then (ds/dt) = (dd/dt)N + d(dN/dt) < 0; since (dN/dt) > 0 then (dd/dt) < 0.

A similar statement can also be made for the "slow" evolution when the inflow q is slowly growing, but only for "ordered" systems, in which di > d2 > d3. Only in this case >d/>q < 0 (in the first interval, when 0 < q # did2/«i, d = di = const), i.e. the value of d decreases with the growth of q. If the "order" was violated (as occurs, for instance, in the

Length 1 Length 2 Length 3

Length 1 Length 2 Length 3

Fig. 7.5. "Slow" evolution of the total biomass (N) and the mean value of the specific entropy production (d) as a function of inflow q: a0 = a1 = 1; d1 = 3, d2 = 1, d3 = 2.

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